A Simple Calculus for Discrete Systems, Part B
Mathematical model for man machine development cycles
Leonard, J. N.↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Singular extremaloids in optimal control theory and the calculus of variations
Definitions of extremaloids in relation to optimal control problems and Lagrange problems based on Pontryagin principles
Straeter, T. A.↗
Representation theory of SO/4, 1/ and E/3, 1/ - An explicit spinor calculus
Explicit spinor basis states for irreducible unitary representations of spinor covering group of deSitter group SO/4,1/
Holman, W. J., III↗
Bounded state variables and the calculus of variations
An optimal control problem with bounded state variables is transformed into a Lagrange problem by means of differentiable mappings which take some Euclidean space onto the control and state regions. Whereas all such mappings lead to a Lagrange problem, it is shown that only those which are defined as acceptable pairs of transformations are suitable in the sense that solutions to the transformed Lagrange problem will lead to solutions to the original bounded state problem and vice versa. In particular, an acceptable pair of transformations is exhibited for the case when the control and state regions are right parallelepipeds. Finally, a description of the necessary conditions for the bounded state problem which were obtained by this method is given.
Hanafy, L. M.↗
Theory and calculus of cubical complexes
Combination switching networks with multiple outputs may be represented by Boolean functions. Report has been prepared which describes derivation and use of extraction algorithm that may be adapted to simplification of such simultaneous Boolean functions.
Perlman, M.↗
Calculus of variations and mathematical theory of optimal control
A bibliography is presented of publications and reports resulting from the research project.
Sagan, H.↗
Development of Boolean calculus and its application
Formal procedures for synthesis of asynchronous sequential system using commercially available edge-sensitive flip-flops are developed. Boolean differential is defined. The exact number of compatible integrals of a Boolean differential were calculated.
Tapia, M. A.↗
Tables for the numerical determination of the Fourier transform of a function of time and the inverse Fourier transform of a function of frequency, with some applications to operational calculus methods
Explore the source record for details and available documents.
Huss, Carl R↗
Fractional-order Viscoelasticity (FOV): Constitutive Development Using the Fractional Calculus: First Annual Report
This is the first annual report to the U.S. Army Medical Research and Material Command for the three year project "Advanced Soft Tissue Modeling for Telemedicine and Surgical Simulation" supported by grant No. DAMD17-01-1-0673 to The Cleveland Clinic Foundation, to which the NASA Glenn Research Center is a subcontractor through Space Act Agreement SAA 3-445. The objective of this report is to extend popular one-dimensional (1D) fractional-order viscoelastic (FOV) materials models into their three-dimensional (3D) equivalents for finitely deforming continua, and to provide numerical algorithms for their solution.
Freed, Alan↗
Algorithms for the Fractional Calculus: A Selection of Numerical Methods
Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.
Diethelm, K.↗
Toward Question-Asking Machines: The Logic of Questions and the Inquiry Calculus
For over a century, the study of logic has focused on the algebra of logical statements. This work, first performed by George Boole, has led to the development of modern computers, and was shown by Richard T. Cox to be the foundation of Bayesian inference. Meanwhile the logic of questions has been much neglected. For our computing machines to be truly intelligent, they need to be able to ask relevant questions. In this paper I will show how the Boolean lattice of logical statements gives rise to the free distributive lattice of questions thus defining their algebra. Furthermore, there exists a quantity analogous to probability, called relevance, which quantifies the degree to which one question answers another. I will show that relevance is not only a natural generalization of information theory, but also forms its foundation.
Knuth,Kevin H.↗
Quad-Tree Visual-Calculus Analysis of Satellite Coverage
An improved method of analysis of coverage of areas of the Earth by a constellation of radio-communication or scientific-observation satellites has been developed. This method is intended to supplant an older method in which the global-coverage-analysis problem is solved from a ground-to-satellite perspective. The present method provides for rapid and efficient analysis. This method is derived from a satellite-to-ground perspective and involves a unique combination of two techniques for multiresolution representation of map features on the surface of a sphere.
Lo, Martin W.↗
Growth-Adapted Tensegrity Structures: A New Calculus for the Space Economy
We describe a novel approach to create and engineer an economically viable space habitat development technology, for deployment of a lightweight tensegrity habitat structure orbiting at Earth-Moon L2, where onboard robotic assets will use space-based materials to provide water for shielding, irrigation and life support, soil for ecosystem development, and to enable structural maintenance and enhancement. The habitat can become a tourist destination, an economic hub, and a multi-purpose research and support facility for lunar surface development and space ecosystem life sciences.
Materials↗
Fortran mimetic abstraction language (Formal) v0.1.
The Fortran mimetic abstraction language ("Formal") is a domain-specific language (DSL) embedded in Fortran 202Y [1]. Formal provides novel software abstractions for simulating phenomena governed by the partial differential equations (PDEs) of vector and tensor calculus. Such equations model an extremely broad set of physical phenomena, ranging from atmospheric winds to light propagation. Formal's data structures and algorithms mimic in form and behavior continuous functions and operators. Formal supports these mathematical constructs using mimetic discretizations that define a discrete calculus satisfying various tensor calculus theorems, thereby ensuring high-fidelity representations of the physics being modeled. [2] Formal 0.1.0 also lays a foundation for the future use of Fortran 202Y type-safe templates to facilitate the formal verification of tensor contractions in computational physics and artificial intelligence [3]. [1] "Fortran 202Y" is Fortran standard committee's informal designation for the next Fortran revision, which will likely be "Fortran 2028". [2] Corbino, J. and Castillo, J. (2020) Journal of Computational and Applied Mathematics, https://doi.org/10.1016/j.cam.2019.06.042. [3] Haveraaen, M., Järvi, J., & Rouson, D. (2019). Reflecting on Generics for Fortran. https://j3-fortran.org/doc/year/19/19-188.pdf.
Rouson, Damian [Lawrence Berkeley National Laborat↗
Compositional Reasoning for Hierarchical State Machines
Harel statecharts and its derivatives are popular graphical languages for specifying discrete control systems via hierarchical state machines. Separately, there has been a long line of work on specifying concurrent systems with process calculi which come equipped with an algebraic theory, the ability reason compositionally about various temporal properties, and strong type systems. While these two approaches to modeling systems are tantalizingly similar, the integrated reasoning principles that exist for process calculi have not been demonstrated in hierarchical state machines. A key issue is that operational theories for process calculi do not behave like control systems, and thus, there is virtually no tool support for modeling control systems with such languages. For a control system designer, bringing the integrated, more scalable reasoning from the process calculi to state-machine languages would enable the specification of more complex systems and a more modular systems development process. Our insight is that we can recover many important results from the process calculi in hierarchical state machines with local scope. We employ a structural operational semantics, which is ubiquitous in process and 𝜆-calculi but uncommon in hierarchical statemachine formalizations, to enable inductive reasoning about behavior. Taking inspiration from the structure of process calculi metatheories, we define a calculus of refinement and equivalence that we prove sound with respect to local notion of (bi)simulation. Furthermore, we prove that the calculus preserves the behavioral properties of reactivity, observational determinism, traces, and linear temporal properties. Our results are mechanized in the Rocq proof assistant.
97 MATHEMATICS AND COMPUTING↗
Towards a Verifiable Domain-Specific Language for Hardware-Accelerated Stencils
Defining a domain-specific language (DSL) that supports vector-calculus abstractions eases the porting of partial differential equation (PDE) solvers to specialized architectures. Sufficiently high-level abstractions empower users to express universal laws with sufficient generality that the laws must always hold true within their domain of validity. A broad class of PDE solvers employs stencil-based algorithms, the target domain of Berkeley Lab's stencil accelerator chip co-design project. First released as open-source in January 2026, the Formal software framework lays a foundation for defining an embedded DSL based on composable operators that implement mimetic numerical methods -- stencil algorithms that guarantee satisfaction of discrete versions of important vector calculus theorems. The Formal DSL will be the frontend to a new class of stencil-PDE accelerators developed jointly by LBNL, UHCL, and UC Berkeley through the DOE Competitive Portfolios for Computer Science Project. This offers the potential of an order of magnitude acceleration for this important category of computational methods to serve the DOE mission. Future work on the Formal DSL will facilitate software verification via type-safe templates that enable problem-specific correctness proofs relying upon generic function theory and carefully crafted unit tests.
Rouson, Damian↗
Tree-manipulating systems and Church-Rosser theorems.
Study of a broad class of tree-manipulating systems called subtree replacement systems. The use of this framework is illustrated by general theorems analogous to the Church-Rosser theorem and by applications of these theorems. Sufficient conditions are derived for the Church-Rosser property, and their applications to recursive definitions, the lambda calculus, and parallel programming are discussed. McCarthy's (1963) recursive calculus is extended by allowing a choice between call-by-value and call-by-name. It is shown that recursively defined functions are single-valued despite the nondeterminism of the evaluation algorithm. It is also shown that these functions solve their defining equations in a 'canonical' manner.
Rosen, B. K.↗